Electronic transport in disordered graphene antidot lattice devices

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Electronic transport in disordered graphene antidot lattice devices
Power, Stephen; Jauho, Antti-Pekka
Published in:
Physical Review B
Link to article, DOI:
10.1103/PhysRevB.90.115408
Publication date:
2014
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Citation (APA):
Power, S., & Jauho, A-P. (2014). Electronic transport in disordered graphene antidot lattice devices. Physical
Review B, 90(11), 115408. https://doi.org/10.1103/PhysRevB.90.115408

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PHYSICALREVIEWB90,115408(2014)
Electronictransportindisorderedgrapheneantidotlatticedevices
StephenR.Power* andAntti-PekkaJauho
CenterforNanostructuredGraphene(CNG),DTUNanotech,DepartmentofMicro-andNanotechnology,
TechnicalUniversityofDenmark,DK-2800KongensLyngby,Denmark
(Received1July2014;revisedmanuscriptreceived28August2014;published5September2014)
Nanostructuring of graphene is in part motivated by the requirement to open a gap in the electronic band
structure.Inparticular,aperiodicallyperforatedgraphenesheetintheformofanantidotlatticemayhavesucha
gap.Suchsystemshavebeeninvestigatedwithaviewtowardsapplicationintransistororwaveguidingdevices.
Thedesiredpropertieshavebeenpredictedforatomicallyprecisesystems,butfabricationmethodswillintroduce
significantlevelsofdisorderintheshape,positionandedgeconfigurationsofindividualantidots.Wecalculate
theelectronictransportpropertiesofawiderangeoffinitegrapheneantidotdevicestodeterminetheeffectof
suchdisordersontheirperformance.Modestgeometricdisorderisseentohaveadetrimentaleffectondevices
containingsmall,tightlypackedantidots,whichhaveoptimalperformanceinpristinelattices.Largerantidots
displayarangeofeffectswhichstronglydependontheiredgegeometry.Antidotsystemswitharmchairedges
areseentohaveafarmorerobusttransportgapthanthosecomposedfromzigzagormixededgeantidots.The
role of disorder in waveguide geometries is slightly different and can enhance performance by extending the
energyrangeoverwhichwaveguidingbehaviorisobserved.
DOI:10.1103/PhysRevB.90.115408 PACSnumber(s): 72.80.Vp,73.21.Cd,73.23.−b,72.10.Fk
I. INTRODUCTION suggestsapplicationinelectronwaveguiding[19],inanalogy
with photonic crystals where antidot lattice geometries have
Much recent effort in graphene research has focused
alsobeenconsidered[44].
on attempts to introduce a band gap into the otherwise
However, many of these potential device applications are
semimetallic electronic band structure of graphene. Such a
predicated onatomicallyprecisegraphene antidot structures,
feature would allow the integration of graphene, with its
whereas experimental fabrication (primarily involving block
many superlative physical, electronic, thermal, and optical
copolymer etching [29–32] or electron beam lithography
properties, into a wide range of device applications [1,2]. In
[33–38] techniques) will inevitably introduce a degree of
particular, the presence of a band gap is a vital step in the
imperfection and disorder into the system. It may, however,
development of a graphene transistor capable of competing
be possible to control the antidot edge geometries to some
withstandardsemiconductor-baseddevices.
extent by heat treatment [38,45] or selective etching [37,46].
Initialinvestigationsintogappedgraphenewereprimarily
Much like the properties of nanoribbons were found to be
basedaroundgraphenenanoribbons,withtheelectronconfine-
greatly affected by disorder [47,48], recent studies suggest
mentinducedbythepresenceofcrystallineedgespredictedto
thattheelectronicandopticalpropertiesofGALsmayalsobe
introduce a band gap similar to that found in many carbon
stronglyperturbed[21,22,24,27].Weshouldthereforeexpect
nanotubes [3–6]. More recent efforts have turned towards
thatthetransportpropertiesanddevicefidelityofthesystems
graphene superlattices, where the imposition of a periodic
describedabovewilldependonthedegreeofdisorderpresent
perturbation of the graphene sheet is also predicted to open
intheantidotlattice.
upabandgap.Theperiodicperforationofagraphenesheet,to
Motivatedbythisconcernwehavesimulatedawiderange
formaso-calledgrapheneantidotlattice(GAL)ornanomesh,
of finite GAL devices, in both simple-barrier and waveguide
is one such implementation of the latter technique [7–38].
geometries,withvariousdisordertypesandstrengths.Wefind
Periodic gating [39,40] and strain [41,42] are other possible
that the geometries predicted to give the largest band gaps,
routesthathavebeensuggested.
namelythosewithadensearrayofsmallholes,areparticularly
TheoreticalstudiesofGAL-basedsystemshavesuggested
susceptible to the effects of disorder and that transport gaps
that the band-gap behavior in many cases follows a simple
arequicklyquenchedasleakagechannelsformatenergiesin
scaling law relating the period of the perturbation and the
the band gap. Geometric disorder, consisting of fluctuations
antidot size [7]. In other cases, more complex symmetry
in the positions and sizes of the antidots, is found to have
arguments involving the lattice geometry [15,16,23] or the
a particularly dramatic effect. However, for larger antidots,
effect of edge states [11,17,25] can be employed to predict
the signatures of such disorders are found to be strongly
thepresenceandmagnitudeofthebandgap.Furthermore,itis
dependentontheedgegeometryoftheindividualantidots.We
predictedthatonlyasmallnumberofantidotrowsarerequired
observedifferentbehaviorwhentheantidotedgeatomshave
to induce bulklike transport gaps [17,18,24], suggesting the
armchairorzigzagconfigurations,oralternatingsequencesof
useofGALsinfinitebarriersystemswhichdonotsufferfrom
both. The remainder of the paper is organized as follows.
theKlein-tunnellingdrivenbarrierleakageexpectedforgated
Section II introduces the geometry of the systems under
systems [43]. Indeed, the potential barrier efficacy of GALs
consideration,thedetailsoftheelectronicmodelandtransport
calculations and the types of disorder that will be applied.
SectionIIIexaminesindetailthetransportpropertiesoffinite-
*[email protected] lengthbarriergeometrydevices,startingwithpristinesystems
1098-0121/2014/90(11)/115408(15) 115408-1 ©2014AmericanPhysicalSociety
STEPHENR.POWERANDANTTI-PEKKAJAUHO PHYSICALREVIEWB90,115408(2014)
(Sec. IIIA). Small antidot systems with edge roughness zigzag (Z), or armchair (A) geometry. The electronic band
(Sec.IIIB)andgeometricdisorders(Sec.IIIC)arepresented, gap of a triangular lattice of such antidots is predicted [7]
beforeweconsiderlargerantidotsystemsandedgegeometry to scale as E ∼ R. Figure 1(b) shows the atomic structure
G L2
effectswithandwithoutthepresenceofdisorder(Sec.IIID). of a unit cell of the {7,3.0} antidot lattice, which will be
C
SectionIVconsidersdeviceswithwaveguidegeometriesand considered throughout much of this work. The geometry of
with the various types of antidots and disorders discussed hexagonal antidots will be discussed later in the paper. The
previously for barrier devices. Finally, Sec. V summarizes rectangularunit√cellscomposingthedeviceregionhavelength
anddiscussestheimplicationsofourresultsinthecontextof 3La andwidth 3La andcontain12L2 carbonatomsinthe
deviceoptimizationandrecentexperimentalprogress. absence of an antidot (R =0). L is always an integer, but
R can take noninteger values. The semi-infinite leads in the
system are wide zigzag graphene nanoribbons (ZGNRs) and
II. MODEL in the R =0 case the entire device can be considered as a
A. Antidotanddevicegeometry pristine 2LWR-ZGNR, using the usual nomenclature where
the integer counts the number of zigzag chains across the
Thesystemsweconsiderinthisworkconsistoffinite-length
ribbonwidth.
device regions connected to semi-infinite leads. A schematic
geometry of a GAL barrier system is presented in Fig. 1(a).
Thedeviceregionisbuiltfromrectangularcells(blue-dashed B. Electronicandtransportcalculations
lines) containing two possible antidot sites from a triangular The electronic structure of the graphene systems inves-
GAL. This lattice geometry is selected as it is predicted to tigated is described by a single π-orbital nearest-neighbor
alwaysopenanelectronicbandgap,unlikesquareandrotated tight-bindingHamiltonian
triangular lattices where particular unit cell sizes may give (cid:2)
rise to gapless systems [15]. The rectangular cell is exactly H = tijcˆi†cˆj, (1)
twice the size of the hexagonal unit cell of such a lattice, (cid:4)ij(cid:5)
which is shown by red dashed lines. The device length D
L where the sum is taken over nearest-neighbor sites only.
and ribbon width W shown in Fig. 1(a) are given in units
R The only nonzero element, in the absence of disorder, is
of the rectangular cell’s length and width, respectively. The t =−2.7eVfornearest-neighborsites.Throughoutthiswork,
antidotlatticesconsideredarecharacterizedbythesize,shape, we will use |t| as the unit of energy. Carbon atoms can be
and spacing of the antidots. We use the notation {L,R} ,
x removedfromtheantidotregionsbyremovingtheassociated
where L and R are the side length of the unit cell and
rows and columns from the system Hamiltonian. Carbon
antidotradius,respectively,bothgiveninunitsofthegraphene
atoms with only a single remaining neighboring atom are
lattice parameter a =2.46A˚. x =C,Z,A specifies whether
also removed from the system. Any dangling sigma bonds
theantidothasacircularshape(C),orahexagonalshapewith
foracarbonatomwithonlytwoneighboringcarbonatomsare
assumedtobepassivatedwithHydrogenatomssothattheπ
bandsareunaffected.
TransportquantitiesarecalculatedusingrecursiveGreen’s
function techniques. The device region is decomposed into
a series of chains, which are connected from left to right
tocalculatetheGreen’sfunctionsrequired.Thesemi-infinite
leadsareconstructedusinganefficientdecimationprocedure,
which takes advantage of system periodicity [49]. A general
(a) (b) overview of these techniques applied to graphene systems
is given in Ref. [50]. The zero-temperature conductance is
given by the Landauer formula [51] G= 2e2T, where the
h
transmissioncoefficientiscalculatedusingtheCaroliformula
T(E)=Tr[Gr(E)(cid:2) (E)Ga(E)(cid:2) (E)], (2)
R L
where(cid:2) (E)(i =L,R)arethelevelwidthmatricesdescribing
i
thecouplingofthedeviceregiontotheleftandrightleadsand
(c) Edge roughness (d)Position (e)Radial Gr/a is the retarded/advanced Green’s function of the device
regionconnectedtotheleads.TheGreen’sfunctionsrequired
FIG.1.(Coloronline) (a)Schematicofasimpleantidotbarrier
forthiscalculationareacquiredfromasinglerecursivesweep
deviceinagraphenenanoribbon.Thewhitecirclesrepresentregions
through the device. When studying disordered systems it is
where carbon atoms have been removed from the lattice. The
usuallynecessarytotakeaconfigurationalaverageovermany
red and blue dashed lines show the single-antidot and double-
instancesofaparticularsetofdisorderparametersinorderto
antidotunitcells,respectively.(b)Theatomicstructureforasingle
circular{7,3.0} antidot.Schematicsof(c)antidotedgeroughness, discerntheoveralltrend.
C
(d)position,and(e)radialdisorders.Thedashedlinesineachcase In studying the antidot devices in this work, we will also
represent the edge of the pristine antidot. The blue (red) circles in examinelocalelectronicpropertiesinordertounderstandhow
(c)representcarbonatomswithanaddedpositive(negative)onsite transportthroughadeviceisaffectedbyantidotgeometryand
potentialtermproportionaltothecirclesize. disorder.Inparticular,wewillmapthelocaldensityofstates
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ELECTRONICTRANSPORTINDISORDEREDGRAPHENE ... PHYSICALREVIEWB90,115408(2014)
atasitei works [21,22], but here we focus entirely on the effects of
1 antidotdisorder.
ρ (E )=− Im[G (E )] (3)
i F ii F
π
andthebondcurrentflowingbetweensitesi andj III. FINITELENGTHGALBARRIERS
1 A. Cleanbarriers
J (E )=− H Im[Gr(cid:2) Ga] , (4)
ij F (cid:2) ij L ij We begin by examining systems built from the {7,3.0}
C
where H is the relevant matrix element of the Hamiltonian antidotlatticeshowninFig.1(c).Latticescomposedofsuch
ij
in Eq. (1). In larger systems, some spatial averaging of small, tightly-packed antidots have been the focus of recent
these quantities will be performed to ease visualisation. The theoreticalinvestigationduetothelargeelectronicbandgaps,
Green’s functions required for these calculations are more which follow due to the scaling law discussed earlier [7]. A
computationally expensive than those for the transmission recentstudysuggeststhatonlyveryfewrowsofantidotsare
calculation, and involve sweeping forward and back through required to achieve a transport gap in a finite GAL barrier
thedeviceregionusingrecursivemethods[50]. whichisidenticaltothebandgapofthecorrespondinginfinite
GALsystem[17].Thisbehaviorisconfirmedforribbon-based
deviceswithpristineantidotsinFig.2,wherewecomparethe
C. Modellingdisorder
transmissionthroughinfiniteGALribbons(wheretheleadand
Inthiswork,wewillconsider threetypes ofdisorderthat deviceregionsbothcontainidenticalantidotarrays)andfinite
areverydifficulttoavoidduringtheexperimentalfabrication GALbarriers(whereafiniteGALribbondeviceisconnected
of GAL devices. Every circular antidot i in a device is
characterized by three parameters (x ,y ,r ) which, represent
i i i
thex andy coordinatesoftheantidotcenter,andtheantidot
radius, respectively. Hexagonal antidots are parameterized
slightlydifferentlyandarediscussedlater.
The first type of disorder we consider is edge roughness.
ThistypeofdisorderplacesAnderson-likerandompotentials
onsiteswithinasmalladditionalradiusoftheantidotedgeand
mimics,forexample,minoratomicrealignment,randomedge
passivations[27],vacancies[24],orrandomlyadsorbedatoms
nearanantidotedge.Itischaracterizedbytwoparameters,an
additionalradiusδR andastrengthS.Allremainingcarbon
dis
sites in the ring between r and r +δR around an antidot
i i dis
centeraregivenarandomonsitepotentialintherange[−S,S],
asillustratedinFig.1(c).
We also consider more serious geometric disorder in the
formoffluctuationsoftheantidotpositionandradius[21,22].
Antidotpositiondisorderischaracterizedbyδ suchthatthe
xy
antidotsitecoordinatesarechosenfromtheranges
x0−δ <x <x0+δ ,
i xy i i xy
y0−δ <y <y0+δ ,
i xy i i xy
where(x0,y0)arethecoordinatesofantidotiinapristineGAL
i i
system.Antidotradialdisorderallowsforasimilarfluctuation
intheradiusoftheantidotsothattheradiusofeachantidotis
intherange
R−δr <ri <R+δr. FIG.2.(Coloronline) (Top) Transmission through infinite and
finite {7,3.0} GAL barriers. The black dashed curve shows the
These theoretical disorders represent the deviations from C
transmission for the infinite GAL ribbon of width W =10. The
atomicallyprecisesystemsthataremostlikelytooccurduring R
solidredcurveshowthetransmissionthroughasystemwithclean
experimentalfabricationofantidots—namely,thedifficulties
graphene leads and a device length D =4. The grey shaded area
inkeepingantidotedgesperfectlycleanandincontrollingthe L
showsthe(arbitarilyscaled)DOSofthecorrespondinginfiniteGAL
position and size of each antidot with atomic precision. We
sheet.TheinsetshowsthetransmissionasafunctionofD calculated
L
donotaccountfordisorderawayfromtheantidottedregions
at different energies shown by the corresponding symbols on the
and assume that the graphene sheet into which the antidots
energy axis of the main plot. The transmissions in the inset have
are patterned is otherwise defect free. We thus neglect, for beenaveragedoveranarrowenergywindowasdiscussedinthemain
example, defects or adsorbants that may occur in rest of the text.(Bottom)LDOSmaps(blueshading)andcurrentdistributions
graphene sheet due to the various etching procedures. The (orangearrows)throughthefinitedeviceatE =0.1|t|(left)inthe
F
electronic and optical properties of graphene antidots with GALbandgapandE =0.2|t|(right)intheconductingrangeofthe
F
disorderinthegrapheneregionshavebeenconsideredinother GAL.
115408-3
STEPHENR.POWERANDANTTI-PEKKAJAUHO PHYSICALREVIEWB90,115408(2014)
topristineGNRleads).Allthedeviceregionsinthissectionare
W =10unitcellswide,whichcorrespondstoa140-ZGNR
R
orawidthofapproximately30nm.Thedevicelength,D ,is
L
anintegernumberofunitcellwidthsandforthe{7,3.0} case
C
eachoftheseisapproximately5nm.
FortheinfiniteGALribbon,thetransmissioncurve(dashed,
black line) clearly shows a gap corresponding to that in the
DOS of the corresponding infinite GAL sheet, illustrated by
thegreyshadedarea.Outsidethegap,thetransmissiontakes
a series of integer values, as expected by the finite-width,
periodicnatureofthesystem.Theseintegervaluescorrespond
to the number of quasi-one-dimensional channels available
at a given energy. The solid, red curve corresponds to the
transmissionthroughasystemwithcleangrapheneleadsand
D =4, where we note the persistence of the transport gap
L
observedfortheinfinitecase.Thisbehaviorisclearalsointhe
LDOS and current maps presented in the bottom left panel
for an energy in the band gap (E =0.1|t|). The LDOS,
F
shown by the blue shading, quickly vanishes when we move
intotheantidotregionandtherearenochannelsavailablefor
transmission. Outside the gap, we note that the transmission
takesapproximatelyhalfthevalueoftheinfinitecaseandthis
reduction intransmissionemerges fromscattering atthe two
interfacesbetweenthedeviceandcleangrapheneleads.These
interfaces are not present in the infinite GAL ribbon case.
Interferenceeffectsleadtoadditionaloscillationsforthefinite FIG.3.(Coloronline) EffectofantidotedgedisorderonD =4
L
width transmission and the oscillation frequency increases {7,3.0} GALbarriers.Thedashedblackcurveplotsthetransmission
C
with D . These are visible also in the nonuniform LDOS
L through a pristine GAL barrier, whereas the red (blue) curves
distributionshownforEF =0.2|t|inthebottomrightpanel. represent transmission through the same system but with δRdis=
Thetransmissionthroughthebarrierisevidentfromthebond a antidot edge disorder of strength S =0.5|t| (S =|t|) applied.
currentmapshownbytheorangearrows.Theinsetofthetop The lighter, dotted curves corresponds to a single configuration
panelplotsthetransmissioninlogscaleasafunctionofdevice whereas the heavier curves corresponds to averages over 100 such
length D for a number of different energies. In each case, configurations. The grey shading shows the rescaled DOS of the
L
thetransmissionisaveragedoveranarrowenergywindowof corresponding infinite GAL sheet. The insets show the effect of
width0.01|t|centeredonthecorrespondingsymbolshownon alteringthedevicelengthforthesameenergyrangesconsideredin
theenergyaxisofthemainplot.Weconsidertwoenergyvalues Fig.2,wherenowthebottom(top)panelcorrespondstoS =0.5|t|
infirstgap(blackcircleandmagentadiamond),twovaluesin (S =|t|).ThebottompanelsagainshowLDOSandcurrentmapsat
theconductingregion(redxandgreentriangle),andonevalue twoenergiesasinFig.2,butforaspecificinstanceofS =|t|edge
inthesecondgap(bluesquare).Foreachenergyconsidered, disorder.
wenoteaquickconvergenceofthetransmissionasthedevice
lengthisincreased.However,thenumberofunitcellsrequired
for convergence at band gap energies increases with energy.
This suggests that higher-order gaps present in GAL band trend for this type of disorder. In this case, most of the
structures may be more difficult to achieve in finite systems, transport gap has been preserved while the transmission
even before consideration of the disorder effects which we in the conducting region is considerably reduced compared
shalladdressnext. to the pristine system. The gap size is reduced slightly by a
broadeningoftheconductingregion.Thedependenceofthese
features on the device length is demonstrated in the bottom
B. Antidotedgedisorder
paneloftheinset,where,similartoFig.2,thetransmissions
The first disorder type we consider is the Anderson-like are plotted as a function of D for the same energy ranges.
L
antidot edge roughness discussed in Sec. IIC. The main For the energies in the first gap (circles and diamonds) we
panel in Fig. 3 shows transmissions through a D =4 GAL note the robustness of the transport gap and only extremely
L
barrier, where the pristine {7,3.0} antidot case is shown minor fluctuations of the miniscule transmission. For the
C
by the dashed black curve. The red curves correspond to energiesinthefirstconductingrange,theappearanceofalinear
transmission where an edge roughness of radius δR =a decrease in transmissioninthis logarithmic plotcorresponds
dis
and strength S =0.5|t| has been applied. The light, dotted to an exponential decay with device length and hence the
curve represents the transmission for a single configuration introductionoflocalisationfeatures.Interestingly,thistypeof
and the heavier dashed curve is the average over 100 such behavior is also seen for the blue squares corresponding to
configurations. The configurational averaging removes the energiesinthesecondgapofthebulkGAL.Thissuggeststhat
configuration-dependent oscillations and reveals the overall theintroductionofdisorderestablishesanonzeroconductance
115408-4
ELECTRONICTRANSPORTINDISORDEREDGRAPHENE ... PHYSICALREVIEWB90,115408(2014)
inthisenergyrangefornarrowbarriers,whichthendecaysdue curves corresponding to energies in the first (black circles,
tolocalizationeffectsasthebarrierwidthisincreased. magenta diamonds) and second (blue squares) band gaps of
ThebluecurvesinthemainpanelofFig.3correspondto the pristine system are more complex. Here the system is
singleconfigurationandaveragedtransmissionsforaslightly initially insulating, but a finite transmission slowly increases
stronger disorder (S =1.0|t|). We note that the features with disorder before reaching a maximum at some critical
discussed for the S =0.5|t| case above are now even more valueofdisorderstrength.Beyondthisvalue,thetransmission
prominent,withsignificantextrabroadeningandquenchingin decreasesinasimilarmannertotheconductingrangeenergies.
theconductingregion.Infact,thebroadeningoftheconducting Thisnonmonotonicbehaviorrepresentsaninterplaybetween
regionhasnowremovedthesecondgapcompletely,andgives two effects. Firstly, increasing the disorder strength leads
risetotransmissionsatenergiesfarfromthebandedgeofthe to more overlap between the finite DOS clusters and a
pristineormilderdisordercases.ThisisconfirmedbytheD largernumberofpossiblepathsthroughthebarrier.However,
L
dependenceplottedintheupperpaneloftheinset.Wenotethat scattering induced by the disorder also acts to reduce the
the magenta diamonds, corresponding to energies in the gap transmission as the device length or scatterer strength is
previously,arenowalsodisplayingsignaturesoflocalization increased.Atenergieswherethepristinesystemisconducting,
effectsandsignificantnonzerotransmissionforshorterdevice showninthebottomrightpanelofFig.3forE =0.2|t|,the
F
lengths. Examining the LDOS map for such a disorder at introductionofdisorderdoesnotintroducenewpathsthrough
E =0.1|t|(bottomleftpanel)showsthatthedensityofstates the device, and only acts to scatter electrons in the existing
F
nolongervanishesuniformlythroughoutthebarrierasinthe channels.
pristine case. Instead, we observe that clusters of finite DOS
existwithinthepanelregion.Breakingtheperfectperiodicity
C. Geometricdisorders
of the antidot lattice leads to the formation of defect states
whose energies can lie in the band gap of the pristine GAL. Until now, we have not significantly altered the geom-
The formation of such states localized at absent antidots in etry of the antidots in the device region. However, even
an otherwise pristine lattice has previously been studied [7]. introducing disorder only at the antidot edges was capable
Similardefectstatesoccuraroundeachdisorderedantidotin of inducing a significant deterioration in the performance
thesystemunderconsiderationhere,andthecouplingbetween of the barrier device. We shall now consider the geometric
suchstatesgivesrisetothelargerclusters.Theoverlapofthese disordersintroducedinSec.IIC,wherethepositionsandsizes
clustersprovidespercolationpathsthroughthedeviceregion of the antidots have random fluctuations. The red curves in
[52],shownbytheorangearrowsmappingtheaveragedbond the top panel of Fig. 5 show the transmission for a single
currents,leadingtofinitetransmissions. instance (dotted) and a configurational average (solid) of
Within the conducting energy range, increasing the edge antidot position disorder, with δ =a. The transmission for
xy
disorder strength for a fixed device length reduces the a single disorder instance contains many peaks, at energies
transmission. This is clearly seen from a comparison of both inside and outside the gap of the pristine system. The
the red and blue plots in the main panel of Fig. 3, and is bottompanelmapstheLDOSandcurrentthroughthedevice
associated with a corresponding decrease in the localization regionfortheenergyhighlightedbythearrowinthetoppanel.
length.Figure4plotstheaveragedtransmissions,forthesame Thispeakcorrespondstoelectronspropagatingthroughalarge
energyrangesconsideredinFigs.2and3,asafunctionofthe connected cluster of finite DOS in the device region to the
disorder strength S. The intuitive behavior of the conducting oppositelead.
range energies is demonstrated by the monotonic decreases Theshape,size,andtransmissionenergiesofsuchclusters
observed in the red (x) and green (triangle) curves. The arestronglydependentonthespecificdisorderconfiguration.
This leads to a smoother curve when the configurational
averageistaken,andthiscurvehasnolargepeaksbutincreases
reasonablysteadilywithFermienergy.Thedefinitionofaband
gapbecomesdifficult,asthelowestenergytransmissionpeak
occurs at different energies for different configurations. The
1
insetofthetoppanelexaminesthelengthdependenceoverthe
n
o
si usualaveragedenergyranges.Wenotethatallenergiesdisplay
s
mi 0.1 exponentialdecaybehavior,andnoneshowtheflatlineseveral
ns ordersofmagnitudeslowerthatwasassociatedwithbandgap
a
Tr energiesforweakedgedisorder.Thissuggeststhatthebarrier
0.01
ispotentiallyleakyforallenergies,butthatreasonableon-off
ratioscouldbeestablishedbytakingadvantageofthesteady
0.001 rise in average transmission with Fermi energy. This results
0 0.5 1 1.5 2 from the greater density of single-configuration transmission
Disorder strength ( | t | )
peaksastheenergyisincreased.Thegeometryofthedevice,
FIG.4.(Coloronline) Averaged transmissions through a and in particular the aspect ratio, may also play a significant
{7,3.0}C GAL barrier as a function of antidot disorder strength role.ForagivendevicelengthDL,theprobabilityofopening
S for energy ranges in the first band gap (black circles, magenta a propagating channel through the device will increase with
diamonds),conductingregion(redx’s,greentriangles)andsecond thedevicewidthWR leadingtoagreaternumberofpeaksin
bandgap(bluesquares)ofthepristineGAL. thetransmission.
115408-5
STEPHENR.POWERANDANTTI-PEKKAJAUHO PHYSICALREVIEWB90,115408(2014)
FIG.5.(Coloronline) (Top) Single configuration (dotted) and FIG.6.(Coloronline) ThesamebarriersetupasFig.5,butnow
averaged(solid)transmissionsthroughaD =6barrierof{7,3.0} with δ =a radial disorder, with the dependence on device length
L C r
antidots with δ =a position disorder. Grey shading shows the in the inset. Grey shading again shows the corresponding pristine
xy
corresponding pristine GAL density of states. The inset shows the GAL density of states. (Bottom) LDOS (blue shading) and current
dependenceondevicelengthfortheusualenergyranges.(Bottom) distribution (orange arrows) through a disordered sample, at the
LDOS (blue shading) and current distribution (orange arrows) energyshownbythearrowinthetoppanel.
through a disordered sample, at the energy shown by the arrow in
thetoppanel.
a low-energy peak, and we note that at this energy there is a
finiteDOSthroughoutmuchofthedeviceregion.
However, the prospects for a usable device composed of The effects of geometric disorder, and radial disorder in
such antidots are dashed further when we examine the case particular, on the band gaps of these small antidot devices
of radial disorder (δr =a) in Fig. 6. We note that the single suggestthatatomiclevelsofprecisionarerequired.Thisplaces
instanceandconfigurationallyaveragedtransmissionsdisplay a huge constraint on experimental fabrication and poses a
radically different behavior from the position disorder cases. severechallengeformassproductionofsuchdevices.
A higher distribution of peaks in the single configuration
transmission is noted at lower energies, and there is less
D. Effectsofantidotsizeandgeometry
variation in peak heights as a function of energy. This leads
to an average that is reasonably flat over the energy range Intheprevioussection,wedemonstratedthedrasticeffect
considered here, which is reflected in the length dependence that geometric disorder has on the transport gap in a GAL
plots which mostly lie on top of each other. Exponential barrier device. We focused on the {7,3.0} system due to
C
decay,signifyinglocalisationbehavior,isagainseenforallthe the large band gap predicted for such a geometry. However,
energyrangesconsidered.Anincreasedtransmissionatlower one drawback of such small, tightly-packed antidots is that
energiesistobeexpectedinasystemwithsomedistribution even the minimum possible fluctuations in antidot radius or
ofsmallerantidots,duetothebandgapscalinglinearlywith positionconstituteasignificantperturbationofthesystem.At
theantidotradiusinpristinelattices.Thesystemalsocontains theatomiclevel,thepositionandradialfluctuationsconsidered
acertaindistributionoflargerantidots,fromwhichwecould abovegiverisetosituationswherethetotalnumberofcarbon
expect some quenching of transmissions at higher energies. atoms added or removed due to disorder constitute ∼10%
Theinterplayofthesetwoeffectsmaypartiallyaccountforthe of the total number of atoms in the pristine device region.
flatteningoftheaveragedtransmissionasafunctionofenergy. Furthermore, small changes in the antidot radius have a
The LDOS and current are mapped in the bottom panel for dramaticeffectontherelativebandgapsofpristineGALs.For
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ELECTRONICTRANSPORTINDISORDEREDGRAPHENE ... PHYSICALREVIEWB90,115408(2014)
example, the pristine GAL composed of the largest allowed The middle row panels of Fig. 7 plot the conductances
antidots in Fig. 6 (R =4a) would have twice the band gap (solidredlines)throughD =10barriersofthethreedifferent
L
expectedforaGALcomposedofthesmallest(R =2a).We types of {21,12.0} antidots, with the barrier setup shown
should thus expect the extreme behavior noted for antidot schematically in the insets. The densities of states of the
latticeswithsuchadistributionofradii. correspondinginfiniteGALsystemsareshownbytheshaded
Latticescomposedoflargerantidotsarethuslesslikelyto gray areas behind the transmission curves. We note that the
beasdramaticallyeffectedbythesameabsolutefluctuationsin simple scaling law proposed in Ref. [7] suggests that these
positionandradius,andarealsomoreexperimentallyfeasible. systems should have a band gap of approximately 0.3eV, so
In this section, we shall consider the {21,12.0} family of that we should expect nonzero conductance features above
antidotlattices.WhereastheR =3.0a antidotwastoosmall ∼0.05|t|. However, for the zigzag and circular cases there
todisplayanysignificantedgefeatures,weseethatthecircular are conductance and DOS peaks at significantly smaller
R =12.0a antidotshowninFig.7(a)displaysanalternating energies. Such features have been observed previously for
sequenceofzigzagandarmchairedgesegments.Toexplorethe similarsystemsandareassociatedwithzigzagedgegeometries
roleofantidotgeometryfurther,wealsoconsiderzigzagand [17,53].Theyarerelatedtothezero-energyDOSpeaksseen
armchairedgedhexagonalantidotsasshownschematicallyin in zigzag nanoribbons, but here they have their degeneracy
panels (b) and (c), respectively. The “R” index in {L,R} brokenandareshiftedawayfromzeroenergyduetohybridiza-
A/Z
referstothesidelengthforhexagonalantidots,sothatzigzag tions between edges around antidots and between the edges
andarmchairedgedantidotsareofasimilarsize,butslightly of neighboring antidots. Indeed, due to alignment between
smaller,thantheircircularcounterpartwiththesameindices. zigzag-edged antidots and the hexagonal unit cell of the
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
FIG.7.(Coloronline) (a)–(c) Unit cell geometries, (d)–(f) transmissions (red lines) through pristine D =10,W =6 barriers, and
L R
(g)–(i)zoomedLDOS(blueshading)andcurrent(orangearrows)mapsectionsforcircular,andzigzagandarmchairedgedhexagonalantidots.
Theinsetsin(d)–(f)showaschematicofthebarrierdevices,withtheredboxeshighlightingtheareasshownin(g)–(i).Thearrowsshowthe
energiesatwhichthemapshavebeencalculated,andtheshadedgreyareasshowthe(rescaled)densitiesofstatesforthecorrespondinginfinite
GALs.
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STEPHENR.POWERANDANTTI-PEKKAJAUHO PHYSICALREVIEWB90,115408(2014)
triangularlatticeusedthroughoutthiswork(andtheunderlying
graphene lattice), we observe that pristine {L,R} systems
Z
essentiallyconsistofanetworkof2(L−R)-ZGNRsmeeting
at triple junctions. The low-energy peak features are present
notonlyinthe{21,12.0} system,butinthe{21,12.0} system
Z C
also due to the alternating sequence of zigzag and armchair
edgesinthecircularantidot.Incontrast,the{21,12.0} system
A
shows no low-energy peaks and by construction contains no
zigzagedgesegmentswiththepossibleexceptionofveryshort
sectionswheretwoarmchairedgesmeet.Thearmchair-edged
antidot is also misaligned with respect to the hexagonal unit
celloftheGAL,sothatthelatticeisnotasimplenetworkof
regular width AGNRs but a more complex system which for
largeantidotsconvergestoanetworkofconnectedtriangular
quantumdots.
Thebottompanelsshowthedensityofstates(blueshading)
and current profile (orange arrows) for a small section (red
FIG.8.(Coloronline) Single instance (black, dashed lines)
rectangleinmiddlepanelinsets)ofeachdevice,attheenergies
and configurationally averaged (red lines) transmissions through
highlighted by the arrows in the middle panels. The circular {21,12.0} barriers composed of circular (left), zigzag (center), and
and zigzag antidot cases both correspond to a low-energy armchair (right) edged antidots with δ =a position disorder (top
xy
peak,andconfirmthattheselow-energytransportchannelsare row) or δ =a size disorder. Grey shading shows the DOS of the
r
mediatedbystatesprincipallylocalizednearzigzaggeometry correspondingpristineGALsheet.
edge segments which are shaded dark blue in the density of
states plots. The current behavior in these two cases is quite
different however due to the formation of circulating current of disorder. In contrast, the {21,12.0} system has a bulk
A
patterns between zigzag segments of neighboring antidots in density of states and barrier transmission properties, shown
the {21,12.0} case. This is in contrast to the more uniform inFig.7(f),whicharemoreconsistentwiththesimplescaling
C
current behavior noted for the zigzag-edged antidots. The law. There are no sharp peaks in the gap region, and the
narrow band of peaks present at slightly higher energies LDOS map and current profiles corresponding to a bulk
(just above (below) 0.05|t| for circular (zigzag) antidots) is conducting energy, reveal a more uniform behavior than the
also quite strongly localized near zigzag edges and gives other antidot types and an absense of edge effects. In the
rise, in the {21,12.0} case, to currents circulating around pristine case, at least it appears that armchair-edged antidots
C
individual antidots. As in the case shown here, the current have a significant advantage over zigzag or mixed edges for
loopsarenotentirelyself-contained,andelectronspropagate applicationsrequiringatransportgap.
between neighboring loops to provide a transport channel The panels in Fig. 8 show the effects of position and size
throughthebarrier.Theselow-energychannelsthroughantidot disorderonD =6barrierdevicesforeachantidotgeometry
L
barriers with zigzag edge segments significantly reduce the consideredintheupperpanelsofFig.7.Wenotethatradial,
effective transport gap of the devices. Furthermore, the first or size, disorder is applied slightly differently to hexagonal
band gap in these devices emerges from the splitting of the antidots. In this case, the fluctuation is applied separately to
zero-energypeakassociatedwithaninfinitezigzagedge.This each edge so that its perpendicular distance from the antidot
is separate to the splitting of these states, which may occur centerismodifiedbyanamountintherange[−δ ,δ ].Thesize
r r
due to electron-electron interactions and which may lead disorder for these antidots thus removes the regularity of the
to spin-polarized zigzag-edged states [25], similar to those hexagonsandresultsinzigzag/armchairsegmentsofvarying
predicted for ZGNRs. A study of spin-polarized systems is lengths within the one antidot. This disorder thus mimics
beyondthescopeofthecurrentwork,butitisworthnotingthat growthortreatmentmethodswhichmayfavourtheformation
forantidotgeometriesconsistingoflongnarrowZGNRs(i.e., ofaparticularedgegeometry,butwithoutnecessarilyresulting
largeLandRvalues)thebandgapispredictedtodependonly in perfectly regular perforations. An example of armchair-
on the ribbon width [25] and the usual band gap scaling law edged GALs with such a disorder can be seen in the upper
[7]nolongerholds.Itisworthnoting,however,thatdisorder panelofFig.9.
is predicted to interfere significantly with the formation of TheredcurveineachpanelofFig.8showsthetransmission
spin-polarizededgestates[54].Thissuggeststhatrobustband averagedover200instancesofdisorderandthedottedblack
gaps as a result of spin splitting may only be a feature of line shows the transmission for a single configuration. The
pristinelatticeswithparticulargeometries. grayshadingoncemoreshowstheDOSforaninfinitepristine
Larger zigzag antidots, which tend to have longer zigzag GAL.Thestrengthofthedisorderisagainδ =aforposition
xy
segments, will have smaller band gaps within our spin- disorder and δ =a for size disorder. The behavior in these
r
unpolarizedmodelduetoweakercouplingwithothernearby plotscanbecomparedtothatobservedforthesamedisorder
zigzag segments. This is in direct opposition to the simple types in smaller antidots shown in the upper of Figs. 5
scalingbehaviorofsmallerantidots,andsuggeststhatzigzag and 6. For all three antidot geometries, the magnitude of
edge segments may be unsuitable for applications involving the transmission is significantly smaller than for the pristine
sizable band gaps even before we consider the introduction cases due to localisation effects. However, the most striking
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ELECTRONICTRANSPORTINDISORDEREDGRAPHENE ... PHYSICALREVIEWB90,115408(2014)
(a) InFig.9,weexaminethetransportpropertiesof{21,12.0}A
barrierswithmuchstrongerlevelsofdisorder.Panel(a)shows
abarriersetupwherebothapositiondisorder(δ =3a)and
xy
a size disorder (δ =3a) have been applied. In addition, an
r
edge roughness disorder (S =|t|) has been applied to atoms
within δR =a of the new antidot edges. The transmission
dis
for this disorder realisation, shown in panel (b) by the black
dottedcurve,mainlyconsistsofaseriesofpeakswhichoccur
abovethebandedgeofthepristineGAL.Theconfigurationally
averaged transmission (solid, red curve) suggests that the
(b) transmission remains near zero throughout much of the
pristinebandgapforeveryconfiguration,beforeconfiguration-
dependentpeaksleadtoafiniteaveragedtransmissionabove
thebandedge.Unlikethemildgeometricdisorders,considered
forzigzagandcircularantidotsearlier,thereisaclearcontrast
betweenthetransmissionbehavioraboveandbelowtheband
edge, and this is visible in both the single and averaged
transmission plots. Panels (c) and (d) show the LDOS and
(c)
current maps for the red-dashed region highlighted in panel
(a), for the energies shown by arrows in panel (b). It is
clear that the barrier is still effective at the lower energy,
and that transmission through the disordered barrier can be
switched on by raising the Fermi energy. In fact, this highly
(d) disordered barrier proves a more robust switch than any of
the circular, zigzag-edged, or {7,3.0} antidot systems with
C
much milder geometric disorder. These results suggest that
fabricationmethods,whichfavourtheformationofarmchair
edgedperforations,arekeytoproducingantidotlatticeswhose
electronicfeaturesarerobustinthepresenceofmildtomedium
FIG.9.(Coloronline) A barrier of {21,12.0} antidots with strengthgeometricdisorder.
A
strong position (δ =3a) and radial (δ =3a) disorders applied,
xy r
in addition to an edge roughness disorder of strength S =|t|. The IV. FINITEGALWAVEGUIDES
pristine case is shown by dashed black lines in (a). The disorder
We now turn our attention to the waveguide geometry,
instanceshownin(a)hasthetransmissionshownbytheblackdotted
where GAL regions surrounding a pristine graphene strip
curvein(b),alongsideanaverage(red)over100suchconfigurations
and the DOS of the pristine {21,12.0} sheet (grey shading). The are employed to confine electron propagation to quasi-one-
A
dimensional channels in the strip. The formation of these
LDOS/currentmapsin(c)and(d),shownforthereddashedregion
in(a),showthatthissystemisstillaneffectivebarrieratenergiesin propagating channels has been investigated previously for
thebandgapofthepristineGAL. infinite waveguides with small, circular antidots and the
band structure of such systems was calculated using both
numerical and analytic, Dirac equation based approaches
feature in these plots is once more the contrast between the [19]. Waveguiding in graphene has also been investigated
armchair-edged hexagonal and the other antidot types. The usinggates,insteadofantidots,todefinethewaveguideedge
single configuration curves for circular and zigzag antidot and induce confinement [55]. Aside from immediate device
barriers contain many peaks throughout both the band gap application, graphene waveguides may provide a platform
andconductingenergiesoftheirpristinecounterparts.Thisis forexploringfundamentalphysicalphenomenalikeCoulomb
becausethesedisorderedsystemscontainpercolatingtransport drag[56].AschematicofafiniteGALwaveguideisshownin
channels between zigzag segments, which occur at energies Fig.10,wherethedevicelengthDLisdefinedasinthebarrier
nearthepristinechannelenergiesshowninFig.7.Averaging case.Thetotalribbonwidthisnow
overmanyconfigurations,wecanseeeitherbroadening(asin W =2W +W ,
R AL G
thezigzagcasewithpositiondisorder)oracompletesmearing
of the pristine features. The picture is very different for the whereWAL countsthenumberofrectangularcellsofantidots
armchaircase,wheredisorderinducesmerelyabroadeningof surrounding WG pristine graphene cells. We also define the
thepristineconductingrange,sothatthebandgapisreduced effectivewidthofthewaveguide,fromRef.[19],as
but not removed. The effect seen here is similar to that of W =W + 1.
mild edge disorder in the smaller antidot systems shown in eff G 2
Fig.3.Thisisnotsurprisingasthelengthscaleofgeometric Thisparameter takesintoaccount thatthepropagation chan-
disorder considered here is small compared to the antidot nels through an infinite pristine waveguide do not decay
size and separation so that it behaves effectively as an edge immediatelyattheedgeoftherectangularunitcell,butsurvive
perturbation. a small distance into the antidot region. It has been used to
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